Am I the only one having difficulty with the Sfard article? I have always found the concept of "mathematical objects" to be challenging. How do we clearly define a mathematical object? CZ 3/8
Let me pose the question mentioned on page 43 of the Sfard article: "what should come first in the process of learning: the knowledge of mathematical objects, or the use of the symbols representing these objects?" (Is this the mathematical equivalent of the chicken and egg question?) CZ 3/8
-- Christina, I had a similar question, and am curious if this phenomenon that Sfard discusses is parallel or in some way related to what Simon et al. (2004) discussed in terms of the learning paradox. NF 3/8
Voigt (1996) makes the following claim: "especially in introductory situations, we cannot presume that the learner would ascribe specific meanings to the topic by themselves - meanings that are compatible with the mathematical meanings the teacher wants the student to learn" (p. 25). What implications does this statement have on teaching? (JE, 3/8)
-- Jonathan, the quote that you pointed out also reminds me of Voigt's (1996) conjecture that " the negotiation of meaning is a necessary condition of learning if the students’ background knowledge differs from the knowledge the teacher wants the student to gain " (p. 43). I see this as highlighting the importance of taken as shared knowledge and Voigt's premise here that meaning is a social process and conceptualized as a social construction between people (Voigt, 1996, p. 34). It seems that any implications for teaching would depend on whether the theoretical perspective or position of the teacher is consistent with what Voigt is proposing. Or, is this indeed the question you are asking?... NF 3/8
Given our discussions on the forms of constructivism over the past few weeks, do you agree or disagree with Voight's (1996) claim that Piaget was a radical constructivist? Why or why not? (see pp. 25-26) (JE, 3/8)
--Correct me if I'm wrong, but the very term, "radical constructivism" is a term coined by von Glasersfeld to describe Piaget's work and the elaborations of the work by von Glasersfeld. Is it not true that Piaget is a radical constructivist by ~definition~? (RK)
Based on this week's readings, how can we answer the question of what "meaning" is in the mathematics classroom? (JE, 3/8)
From Ernest (1994), I am still confused whether dialogical and dialectical are used interchangeably or they are different. I will be glad to have this understanding. (NA)
A discussion on "what is mathematics" might be interesting in light of Ernst's discussion of this matter in relation to a culture, a language, a social system (e.g., Ernst, 1994, p. 34). NF 3/8
-- Nicole, I would be very interested to hear what other people have to say about this. And I ~promise~ to behave.
I am having a hard time understanding Ernst's distinction between dialogical and dialectical, and in a related vein, his discussion of Lakatos' cycle of logic was quite perplexing (Ernst, 1994, p. 41 and 42, respectively). NF 3/8
I though Sfard's notion of "templates" was compelling. It made me think of Piaget's sensory-motor stage as a time when one builds templates out of which future meanings grow. It also seemed suggestive of a source of intution. That is, intution may lie largely in metaphor which is, as Sfard writes, "the effect of transferring templates from discourse to discourse" (p. 68). For example, in set theory there are a lot basic metaphors referring to position. The basic statements are often along the lines of "x is in A" or "A is contained in B." Basic topological notions are often the same way, "x is in the boundary of E." These may have precise definitions, but (speaking for myself) the definitions are often a way to prove that statements arrived at through intuition are correct. In other words, when I prove that the closure of a set is the union of the interior and boundary, I think first in terms of visual metaphor (I transfer templates built during my everyday discourses of position into the topological discourse). This gives me an idea of how I'd like to approach the proof and then I get out the definitions and go to work. At times of difficulty during the proof I may again refer to these templates of position (dare I say I fold back?!). I'd like to hear more about Sfard's notion of "template." How does it relate to other psychological theories? Are there ways in which her theory of templates is ~not~ a compelling description of mathematical intuition? (I was quite struck by it). (RK, This morning)
Sfard (2000) argues that in AR discourse when referring to concrete objects eliminates difficulties because participants refer to features directly (p. 41). Can mental objects not mediate AR discourse as well? (NA)
I feel badly confused about the difference between a "sign," a "signifier," and the "signified." Perhaps the class would be willing to spend five minutes on the differences to bring the slow kid up to speed. (RK, Later the same morning)
When I was reading Sfard's chapter, I was intrigued by the difference in the word associations made by children and adults (p. 56-57) and, consequently, the differences that this may imply regarding meaning production. When does this "change in association" happen for individuals? Is guiding children to get to "paradigmatic meaning production" one of our essential goals as a teacher? Is this meaning production something we can facilitate or does it occur naturally in individuals? (JH 3/9)
Let me pose the question mentioned on page 43 of the Sfard article: "what should come first in the process of learning: the knowledge of mathematical objects, or the use of the symbols representing these objects?" (Is this the mathematical equivalent of the chicken and egg question?) CZ 3/8
-- Christina, I had a similar question, and am curious if this phenomenon that Sfard discusses is parallel or in some way related to what Simon et al. (2004) discussed in terms of the learning paradox. NF 3/8
Voigt (1996) makes the following claim: "especially in introductory situations, we cannot presume that the learner would ascribe specific meanings to the topic by themselves - meanings that are compatible with the mathematical meanings the teacher wants the student to learn" (p. 25). What implications does this statement have on teaching? (JE, 3/8)
-- Jonathan, the quote that you pointed out also reminds me of Voigt's (1996) conjecture that " the negotiation of meaning is a necessary condition of learning if the students’ background knowledge differs from the knowledge the teacher wants the student to gain " (p. 43). I see this as highlighting the importance of taken as shared knowledge and Voigt's premise here that meaning is a social process and conceptualized as a social construction between people (Voigt, 1996, p. 34). It seems that any implications for teaching would depend on whether the theoretical perspective or position of the teacher is consistent with what Voigt is proposing. Or, is this indeed the question you are asking?... NF 3/8
Given our discussions on the forms of constructivism over the past few weeks, do you agree or disagree with Voight's (1996) claim that Piaget was a radical constructivist? Why or why not? (see pp. 25-26) (JE, 3/8)
--Correct me if I'm wrong, but the very term, "radical constructivism" is a term coined by von Glasersfeld to describe Piaget's work and the elaborations of the work by von Glasersfeld. Is it not true that Piaget is a radical constructivist by ~definition~? (RK)
Based on this week's readings, how can we answer the question of what "meaning" is in the mathematics classroom? (JE, 3/8)
From Ernest (1994), I am still confused whether dialogical and dialectical are used interchangeably or they are different. I will be glad to have this understanding. (NA)
A discussion on "what is mathematics" might be interesting in light of Ernst's discussion of this matter in relation to a culture, a language, a social system (e.g., Ernst, 1994, p. 34). NF 3/8
-- Nicole, I would be very interested to hear what other people have to say about this. And I ~promise~ to behave.
I am having a hard time understanding Ernst's distinction between dialogical and dialectical, and in a related vein, his discussion of Lakatos' cycle of logic was quite perplexing (Ernst, 1994, p. 41 and 42, respectively). NF 3/8
I though Sfard's notion of "templates" was compelling. It made me think of Piaget's sensory-motor stage as a time when one builds templates out of which future meanings grow. It also seemed suggestive of a source of intution. That is, intution may lie largely in metaphor which is, as Sfard writes, "the effect of transferring templates from discourse to discourse" (p. 68). For example, in set theory there are a lot basic metaphors referring to position. The basic statements are often along the lines of "x is in A" or "A is contained in B." Basic topological notions are often the same way, "x is in the boundary of E." These may have precise definitions, but (speaking for myself) the definitions are often a way to prove that statements arrived at through intuition are correct. In other words, when I prove that the closure of a set is the union of the interior and boundary, I think first in terms of visual metaphor (I transfer templates built during my everyday discourses of position into the topological discourse). This gives me an idea of how I'd like to approach the proof and then I get out the definitions and go to work. At times of difficulty during the proof I may again refer to these templates of position (dare I say I fold back?!). I'd like to hear more about Sfard's notion of "template." How does it relate to other psychological theories? Are there ways in which her theory of templates is ~not~ a compelling description of mathematical intuition? (I was quite struck by it). (RK, This morning)
Sfard (2000) argues that in AR discourse when referring to concrete objects eliminates difficulties because participants refer to features directly (p. 41). Can mental objects not mediate AR discourse as well? (NA)
I feel badly confused about the difference between a "sign," a "signifier," and the "signified." Perhaps the class would be willing to spend five minutes on the differences to bring the slow kid up to speed. (RK, Later the same morning)
When I was reading Sfard's chapter, I was intrigued by the difference in the word associations made by children and adults (p. 56-57) and, consequently, the differences that this may imply regarding meaning production. When does this "change in association" happen for individuals? Is guiding children to get to "paradigmatic meaning production" one of our essential goals as a teacher? Is this meaning production something we can facilitate or does it occur naturally in individuals? (JH 3/9)